Polynomial remainder theorem

Summary

In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout)[1] is an application of Euclidean division of polynomials. It states that, for every number any polynomial is the sum of and the product by of a polynomial in of degree less than the degree of In particular, is the remainder of the Euclidean division of by and is a divisor of if and only if [2] a property known as the factor theorem.

Examples edit

Example 1 edit

Let  . Polynomial division of   by   gives the quotient   and the remainder  . Therefore,  .

Example 2 edit

Proof that the polynomial remainder theorem holds for an arbitrary second degree polynomial   by using algebraic manipulation:

 

So,

 
which is exactly the formula of Euclidean division.

The generalization of this proof to any degree is given below in § Direct proof.

Proofs edit

Using Euclidean division edit

The polynomial remainder theorem follows from the theorem of Euclidean division, which, given two polynomials f(x) (the dividend) and g(x) (the divisor), asserts the existence (and the uniqueness) of a quotient Q(x) and a remainder R(x) such that

 

If the divisor is   where r is a constant, then either R(x) = 0 or its degree is zero; in both cases, R(x) is a constant that is independent of x; that is

 

Setting   in this formula, we obtain:

 

Direct proof edit

A constructive proof—that does not involve the existence theorem of Euclidean division—uses the identity

 

If   denotes the large factor in the right-hand side of this identity, and

 

one has

 

(since  ).

Adding   to both sides of this equation, one gets simultaneously the polynomial remainder theorem and the existence part of the theorem of Euclidean division for this specific case.

Applications edit

The polynomial remainder theorem may be used to evaluate   by calculating the remainder,  . Although polynomial long division is more difficult than evaluating the function itself, synthetic division is computationally easier. Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remainder theorem.

The factor theorem is another application of the remainder theorem: if the remainder is zero, then the linear divisor is a factor. Repeated application of the factor theorem may be used to factorize the polynomial.[3]

References edit

  1. ^ Piotr Rudnicki (2004). "Little Bézout Theorem (Factor Theorem)" (PDF). Formalized Mathematics. 12 (1): 49–58.
  2. ^ Larson, Ron (2014), College Algebra, Cengage Learning
  3. ^ Larson, Ron (2011), Precalculus with Limits, Cengage Learning